What is a "logit probability"?
Indeed I haven't seen the term "logit probability" used in many places other than that specific paper. So, I cannot really comment on why they're using that term / where it comes from / if anyone else uses it, but I can confirm that what they mean by "logit probability" is basically the same thing that is more commonly referred to simply as "logits": they are the raw, unbounded scores of which we generally push a vector through a softmax function to generate a discrete probability distribution that nicely adds up to $1$.

This definition fits the one you linked from wikipedia (although that link only covers the binary case, and AlphaGo Zero would have multinomial logits since it has more than two outputs for the policy head).

In the AlphaGo Zero paper, the described architecture has a "linear output layer" (i.e. no activation function for the outputs, or the identity function as activation function for the outputs, or however you like to describe it) for the policy head. This means that these outputs are essentially unbounded, they could be any real number. We know for sure that these outputs cannot directly be interpreted as probabilities, even if this isn't stated quite explicitly in the paper.

By calling them logits (or logit probabilities for reasons unknown to me), they are essentially implying that these outputs will still be post-processed by a softmax to convert them into a vector that can be interpreted as a discrete probability distribution over the actions, even if they do not explicitly describe a softmax layer as being a part of the network.

It is indeed possible that in Leela Zero they decided to make the softmax operation explicitly a part of the Neural Network architecture. Mathematically they end up doing the same thing... the AlphaGo Zero paper implies (by using the word "logit") that a softmax is used as a "post-processing" step, and in Leela Zero they explicitly make it a part of the Neural Network.

Here are a couple more sources for the reasoning that usage of the word "logit" basically implies usage of a softmax, though indeed they do not cover the term "logit probability":


https://stats.stackexchange.com/a/52836/144483
https://stackoverflow.com/a/34243720/6735980